# 2. Ambiguous Rings Based on a Polygon

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This Demonstration explores ambiguous rings.

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Contributed by: Erik Mahieu (May 2018)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The parametric equation of a circular cylinder with radius inclined at an angle from the vertical is given by:

,

with parameters and .

Define the functions:

and

.

The and functions define the composite curve of the -gonal base of the polygonal cylinder [1].

The parametric equation of a polygonal cylinder with sides and radius rotated by an angle around its axis is given by:

,

with parameters and .

To find the equation of the intersection curve, set the corresponding components equal. This gives the three equations:

,

,

,

with

.

These are equations with four variables, , , and . Eliminating , and by solving the equations gives the parametric equation of the intersection curve with θ as the only parameter:

.

Reference

[1] E. Chicurel-Uziel, "Single Equation without Inequalities to Represent a Composite Curve," *Computer Aided Geometric Design*, 21(1), 2004 pp. 23–42. doi:10.1016/j.cagd.2003.07.011.

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